# Light propagating at angle, under Lorentz transformation

1. Nov 25, 2007

### wtronic

well, i have been going over this problem for about 4 hours and everytime i get closer and closer to some answer but I am never convinced i am right.

THE PROBLEM:
let lambda be the wavelength of light propagating in the x-y plane at an angle theta with respect to the x-axis in the (x,y,z,t) frame. What is its wavelength and direction of propagation in the (x',y',z',t') frame.

I know the wave equation for a wave is

$$\Phi$$cosK[x + ct]

but since traveling at an angle, theta should I put?

$$\Phi$$cosk[xcos$$\theta$$ +ct]
+
$$\Phi$$cosk[ycos$$\theta$$ +ct]

and using Lorentz's tranformation

x' = $$\gamma$$(x - vt)
t' = $$\gamma$$(t - ux/c^2)

I would appreciate any help, thanks

2. Nov 26, 2007

### bernhard.rothenstein

You find a full explanation consulting
Robert Resnick, Introduction to Special Relativity John Willey and Sons 1968 pp.84-87.
It is based on the invariance of the phase of a plane electromagnetic wave propagating along an arbitrary direction and makes use of the Lorentz transformations for the space-time coordinates of the same event.
I think that all that can be done involving only the time dilation effect.

3. Nov 26, 2007

### JesseM

This page has the equation for a plane wave traveling at an angle, as does this one. However, it seems to me this is an over-complicated approach--you don't actually need to know the amplitude of the wave at every point in space, you just need the distance between the peaks. So, you could just imagine two particles traveling in a straight line at an angle theta relative to the x-axis, with the distance between the particles equal to the wavelength--if you figure out the equation for each particle's position as a function of time in the first coordinate system, it's not too hard to find the corresponding equations in the second coordinate system (especially because you can just pick two points on each particle's worldline and translate their coordinates individually).

4. Nov 27, 2007

### wtronic

thanks for the help, I think I solved the problem but i am not confident it is correct. I will just turn it into my professor and see what his coments are.
JesseM- Thanks for the link, I went there and it was a good help; I think the approach I was doing is correct.

thanks for the help